Doubly-curved crystals are commonly used for focusing monochromatic radiation beams, particularly in the X-ray range, and for wavelength dispersion in X-ray spectrometers. To produce such devices, the crystal curvature must be carefully controlled to give the desired focusing properties. Exemplary methods for forming doubly-curved crystals of this sort are described in U.S. Pat. Nos. 4,807,268, 4,780,899, 4,949,367, 6,236,710 and 6,498,830, whose disclosures are incorporated herein by reference.
When a thin film is deposited on a substrate, compressive or tensile stresses may be created in the film, depending on the conditions of deposition. These stresses cause tensile or compressive internal forces in the substrate/thin film assembly, which may cause bending moments in the assembly. Hoffman et al. studied and reported on these stress phenomena in an article entitled, “Internal Stresses in Cr, Mo, Ta, and Pt Films Deposited by Sputtering from a Planar Magnetron Source,” Journal of Vacuum Science and Technology 20:3 (March, 1982), pages 355-358, which is incorporated herein by reference. The authors found that when the pressure of argon process gas was below a certain level during sputter-deposition of the films, the stresses tended to be compressive.
Shen et al. described the evolution of stresses and the accompanying changes in overall curvature due to patterning of silicon oxide lines on silicon wafers in an article entitled, “Stresses, Curvatures, and Shape Changes Arising from Patterned Lines on Silicon Wafers,” Journal of Applied Physics 80:3 (August, 1996), pages 1388-1398, which is incorporated herein by reference. The authors developed a parametric numerical model for the stresses created in SiO2 lines of different dimensions and used the model to predict the curvature caused by these stresses in silicon wafers on which the lines were deposited.